there are some turbocharged PA 44s around. Might be more useful in the mountainous area.
Fly to rapid city, south dakota, and rent a car there and drive to mt rushmore.
be sure not to land at the air force base by mistake! or HILL afb near salt lake city by mistake.
have you checked logan utah airport for rentals? not too far from SLC.
You might want to check out oxygen use too. And fees.
I've been to Mt Rushmore by car. Interesting but there are better places to see. Not quite as impressive as in the movie, "north by northwest".
Good luck, happy landings. Remember that the PA44 has a single engine ceiling of only about 5000' and that's if you are lucky.
You might also consider not using FULL fuel to improve performance.
Nice little plane though. Really nice ailerons.
REMEMBER the horizon is at the base of the mountains not the tops!
You might want to fly to San Francisco if you have 10 hours to use.